We consider a wave function evolving according to the two-dimensional Schrödinger's equation
subject to an unknown Hamiltonian function.
The unknown potential function is taken to be a finite well in a hexagonal shape.
We collect measurements of the wave function and its velocity
and train an exact DMD model and a piDMD model.
In practice, empirical measurements of the wavefunction can be obtained through optical
tomographic methods [1]}, [2]}
or sequential measurements of complementary variables [3]}.
The data matrices \(X\)  and \(Y\)  are formed from measurements of \(\vert \Psi (t)\rangle \)  and \(i \hbar \frac{d}{dt} \vert \Psi (t)\rangle \)  respectively,
which are themselves formed by superposing eigenfunctions obtained by a finite difference method.
Thus, the matrices \(X\)  and \(Y\)  are connected by an unknown self-adjoint Hamiltonian \(\hat{H}\)  and we may apply our self-adjoint DMD optimization.
The measurements of \(Y\)  are contaminated with Gaussian noise but
the measurements of \(X\)  are clean.
